The phone started as a stock 4.3.3 3GS. I used sn0wbreeze to create a custom 4.3.3 firmware with ipad baseband upgrade and using Tinyumbrella TSS server to sign the firmware. Everything went well however I am getting an error code 37 after iTunes restore. The phone is stuck in DFU mode, I haven't tried to restore it back to any other software after the error 37.

Error 37 is odd. If you're not including boot logo's, it doesn't make sense.

Try building and restoring to 4.1.
You need to download Sn0wbreeze 2.2.1 and the 4.1 ipsw.

Apple is signing 4.1 so Tiny Umbrella is not needed.
If iTunes Error 10xx, use 'exit recovery' from Tiny Umbrella.

Would this be a problem if I tried to upload the ipad baseband the first time around? For some reason when I tried to DFU restore (custom 4.3.3) through itunes, it seemed like it went through however it went right back to DFU without any error codes.

4.1 works. It is very odd 4.1 works and nothing else does.
I have a suspicion there's a problem with your SHSH blobs.
Go back to 4.1, install iSHSHit via Cydia get blobs that way and email to yourself.
Then change the name/extension to allow Tiny Umbrella to use them during the restore.